© 2016 Pearson Education, Inc. Linear Equations in

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© 2016 Pearson Education, Inc. Linear Equations in

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2016 Pearson Education, Inc. Linear Equations in Linear Algebra INTRODUCTION TO LINEAR TRANSFORMATIONS Slide 1.8- 2 2016 Pearson Education, Inc. LINEAR TRANSFORMATIONS Prof Herron likes Slide 1.8- 3 MATRIX TRANSFORMATIONS 2016

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Slide 1.8- 6 MATRIX TRANSFORMATIONS Solution:
Compute

.

Solve for x. That is, solve , or

(1) © 2016 Pearson Education, Inc.

Slide 1.8- 7 MATRIX TRANSFORMATIONS Row reduce the augmented matrix:

(2)

Hence , , and .

The image of this x under T is the given vector b. © 2016 Pearson Education, Inc. ~ ~ ~

Slide 1.8- 9 MATRIX TRANSFORMATIONS To find the answer, row reduce the augmented matrix:

The third equation, , shows that the system is inconsistent.
So c is not in the range of T. © 2016 Pearson Education, Inc. ~ ~ ~

Slide 1.8- 11 SHEAR TRANSFORMATION The key idea is to show that T maps line segments onto line segments and then to check that the corners of the square map onto the vertices of the parallelogram.
For instance, the image of the point is

, © 2016 Pearson Education, Inc.

Slide 1.8- 12 LINEAR TRANSFORMATIONS and the image of is .

T deforms the square as if the top of the square were pushed to the right while the base is held fixed.
Definition: T is a linear transformation if:
for all u, v in the domain of T; and,
for all scalars c and all u in the domain of T. © 2016 Pearson Education, Inc.

Slide 1.8- 14 LINEAR TRANSFORMATIONS and . (4)
for all vectors u, v in the domain of T and all scalars c, d.
Property (3) follows from condition (ii) in the definition, because .
Property (4) requires both (i) and (ii):

If a transformation satisfies (4) for all u, v and c, d, it must be linear.
(Set for preservation of addition, and set for
preservation of scalar multiplication.) © 2016 Pearson Education, Inc.

Slide 1.8- 15 LINEAR TRANSFORMATIONS Repeated application of (4) produces the useful linearity principle:
(5)

In engineering and physics, (5) is referred to as a superposition principle.

Think of v1, …, vp as signals that go into a system and T (v1), …, T (vp) as the responses of that system to the signals. © 2016 Pearson Education, Inc.

© 2016 Pearson Education, Inc. Linear Equations in

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